Evolution of public cooperation in a monitored society with implicated punishment and within-group enforcement

Monitoring with implicated punishment is common in human societies to avert freeriding on common goods. But is it effective in promoting public cooperation? We show that the introduction of monitoring and implicated punishment is indeed effective, as it transforms the public goods game to a coordination game, thus rendering cooperation viable in infinite and finite well-mixed populations. We also show that the addition of within-group enforcement further promotes the evolution of public cooperation. However, although the group size in this context has nonlinear effects on collective action, an intermediate group size is least conductive to cooperative behaviour. This contradicts recent field observations, where an intermediate group size was declared optimal with the conjecture that group-size effects and within-group enforcement are responsible. Our theoretical research thus clarifies key aspects of monitoring with implicated punishment in human societies, and additionally, it reveals fundamental group-size effects that facilitate prosocial collective action.


SI-2000 Maribor, Slovenia
In the Supplementary Information, we first provide additional details regarding the stationary fraction of cooperators in infinite and finite populations ( Fig. S1 and Fig. S2).
Second, we explore the results regarding the stationary distribution and the average cooperation level for large peer punishment cost in finite populations (Fig. S3). Third, we investigate the effects of selection intensity on the stationary distribution and the average cooperation level in finite populations (Fig. S4). Fourth, we study the effects of mutation rates on the stationary distribution and the average cooperation level in finite populations (Fig. S5). Finally, we examine the effects of discounting factor of implicated punishment fine on the internal root of the gradient of selection both in infinite and finite populations ( Fig. S6). Figure S1 shows the stationary fraction of cooperators in infinite populations for different parameter combinations. We find that even if the implicated punishment fine and the within-group enforcement probability are increased greatly, there is still no interior equilibrium for small monitoring probability. But when the monitoring probability increases to a critical value, the interior equilibrium is then present. And the critical value becomes smaller as the implicated punishment fine or the within-group enforcement probability increases. When the monitoring probability further increases, the value of the interior equilibrium decreases monotonically. However, when the parameter variations occur, there is no interior equilibrium for small implicated punishment fine, small within-group enforcement probability, or small group size. When these three parameters ( , , and ) respectively reach the critical values, the interior equilibrium displays, and then its value decreases with increasing these parameters. Accordingly, the basin of attraction of the = 1 steady state is enlarged. In addition, we see that with increasing the monitoring probability , the implicated punishment fine , the within-group enforcement probability , or the group size , the gradient of selection increases in the areas where > 0. But for a fixed value of , , , or , the gradient of selection can always reach the maximal values at an intermediate fraction of cooperators, which is smaller than = 1.

Stationary fraction of cooperators in infinite and finite populations for parameter variations
Furthermore, we compute the interior root of ( ) for finite populations in Fig.   S2, which recovers to that for infinite populations in Fig. S1 when the population size → +∞. We find that the root's value for large is very close to that in infinite    Figure S3 shows the stationary distribution and the average cooperation level in the presence of mutation = 0.01 for large peer punishment cost. We still see that the population spends more time in states where cooperators prevail for larger monitoring probability, larger implicated punishment fine, or larger within-group enforcement probability. While the stationary distribution has a maximum at the full cooperation state

Stationary distribution and average cooperation level for large peer punishment cost
for either a small group size or a large group size. We also observe that the average cooperation level increases with increasing the monitoring probability, the implicated punishment fine, or the within-group enforcement probability, even for a large peer punishment cost. When the monitoring probability, the implicated punishment fine, or the within-group enforcement probability is large, the cooperation level approaches one. But for a larger peer punishment cost, the critical values of these three parameters for a high cooperation level become larger. In addition, with increasing the group size the cooperation level first decreases to a small value which is close to zero, and it is kept   Figure Figure S5 shows the stationary distribution and the average cooperation level for different mutation rates. We see that for small mutation rate (top row), the system spends most of the time near the full cooperation state or the full defection state, leading to maxima of the stationary distribution at = 0 or = . And for large monitoring probability, the stationary distribution will have a maxima at = , which means that cooperation is greatly promoted accordingly. For large mutation rate (second row), we see that the stationary distribution has a maximum at a coexistence state which is close to the full defection state for small monitoring probability. While for intermediate monitoring probability, the stationary distribution keeps a maximum at the coexistence state, but also a second (smaller) maximum at another coexistence state which is close to the full cooperation state. For large monitoring probability, the stationary distribution has only a maximum at a coexistence state which is close to the full cooperation state. These results

Effects of mutation rates on the stationary distribution and the average cooperation level
show that cooperation is promoted for large monitoring probability even for large mutation rate. When the mutation rate is further increased (third row), the stationary distribution only has a maximum centered around the intermediate values of the fraction of cooperators for each value of monitoring probability, and with increasing the monitoring probability the center moves towards the right, corresponding to a larger fraction of cooperators. Similarly, these changes about the stationary distribution occur when we investigate the effects of the implicated punishment fine, the within-group enforcement probability, and the group size for the three different mutation rates. But we find that our main results remain unchanged even at either a very small mutation rate or a very large mutation rate, that is, the system spends most of the time in a state where cooperators prevail for large monitoring probability, for large implicated punishment fine, for large within-group enforcement probability, or for either small or large group size.
In the bottom row, we show the average cooperation level for the three mutation rates. We see that the average cooperation level increases with increasing the monitoring probability, the implicated punishment fine, or the within-group enforcement probability.
While for small mutation rate, it slowly increases with increasing the three parameters. In addition, the cooperation level is highest for small mutation rate when the monitoring probability, the implicated punishment fine, or the within-group enforcement probability is large. While the cooperation level is highest for very large mutation rate when the three parameters are small. We also see that with increasing the group size the cooperation level first decreases, then increases after reaching the minimal value for these three levels of selection intensity.

Effects of discounting factor of implicated punishment fine on the internal root of the gradient of selection
We consider a variant for the implementation of the implicated punishment, and assume that when a group of individuals are punished, each defector incurs a fine , while each cooperator incurs a fine , where (0 < < 1) is a discounting factor of implicated punishment fine. Figure S6 shows the internal root of the gradient of selection as a function of the discounting factor in infinite and finite populations. We find that both in infinite and finite populations the internal root increases with increasing the discounting factor, which means that the basin of attraction of the full cooperation state becomes smaller and cooperators become disadvantageous with increasing the discounting factor.